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32
Computational Geometry Problems in REDLOG
 AUTOMATED DEDUCTION IN GEOMETRY
, 1998
"... We solve algorithmic geometrical problems in real 3space or the real plane arising from applications in the area of cad, computer vision, and motion planning. The problems include parallel and central projection problems, shade and cast shadow problems, reconstruction of objects from images, offset ..."
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Cited by 17 (11 self)
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We solve algorithmic geometrical problems in real 3space or the real plane arising from applications in the area of cad, computer vision, and motion planning. The problems include parallel and central projection problems, shade and cast shadow problems, reconstruction of objects from images, offsets of objects, Voronoi diagrams of a finite families of objects, and collision of moving objects. Our tools are real elimination algorithms implemented in the reduce package redlog. In many cases the problems can be solved uniformly in unspecified parameters. The power of the method is illustrated by examples many of which have been outside the scope of real elimination methods so far.
Approximation by Profile Surfaces
, 1998
"... A new algorithm for approximation of a given surface or scattered points by a surface of revolution is presented. It forms the basis for a study of approximation with prole surfaces. Those are sweeping surfaces traced out by a planar curve when its plane is rolling on a developable surface. Importan ..."
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Cited by 9 (4 self)
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A new algorithm for approximation of a given surface or scattered points by a surface of revolution is presented. It forms the basis for a study of approximation with prole surfaces. Those are sweeping surfaces traced out by a planar curve when its plane is rolling on a developable surface. Important special cases include developable surfaces and pipe surfaces, where the moving curve is a straight line or circle, respectively. The methods are illustrated using applications in reverse engineering of geometric models. keywords: computer aided design, computer aided manufacturing, surface approximation, reverse engineering, surface of revolution, prole surface, pipe surface, developable surface 1 Introduction Recently, the interest in reconstruction of surfaces from point sets such as data from a laser scanner has been increasing, since surface reconstruction possesses a wide range of applications. The motivation for our work partially comes from reverse engineering. Whereas conventio...
The convolution of a paraboloid and a parametrized surface
 Journal for Geometry and Graphics
, 2003
"... We investigate the computation of parametrizations of convolution surfaces of paraboloids and arbitrary parametrized surfaces. In particular it will turn out that the addressed problem is linear such that the convolution of a paraboloid and a rational surface admits rational parametrizations. In add ..."
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Cited by 6 (4 self)
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We investigate the computation of parametrizations of convolution surfaces of paraboloids and arbitrary parametrized surfaces. In particular it will turn out that the addressed problem is linear such that the convolution of a paraboloid and a rational surface admits rational parametrizations. In addition, the convolution of paraboloids and surfaces from special classes, like ruled surfaces or surfaces of rotation will be studied.
Analytic and Algebraic Properties of Canal Surfaces ⋆
"... The envelope of a oneparameter set of spheres with radii r(t) and centers m(t) is a canal surface with m(t) as the spine curve and r(t) as the radii function. This concept is a generalization of the classical notion of an offset of a plane curve. In this paper, we firstly survey the principle geome ..."
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Cited by 5 (0 self)
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The envelope of a oneparameter set of spheres with radii r(t) and centers m(t) is a canal surface with m(t) as the spine curve and r(t) as the radii function. This concept is a generalization of the classical notion of an offset of a plane curve. In this paper, we firstly survey the principle geometric features of canal surfaces. In particular, a sufficient condition of canal surfaces without local selfintersection is presented. Moreover,the Gaussian curvature and a simple expression for the area of canal surfaces are given. We also consider the implicit equation f(x, y, z) = 0 of canal surfaces. In particular, the degree of f(x, y, z) is presented. By using the degree of f(x, y, z), a low boundary of the degree of parametrizations representations of canal surfaces is presented. We also prove the low boundary can be reached in some cases. Key words: Canal surfaces, selfintersections, implicit equations 1.
The intersection of two ringed surfaces and some related problems
 Graphical Models
, 2001
"... We present an efficient and robust algorithm to compute the intersection curve of two ringed surfaces, each being the sweep ∪uC u generated by a moving circle. Given two ringed surfaces ∪uC u 1 and ∪vC v 2, we formulate the condition C u 1 ∩ Cv 2 �=∅(i.e., that the intersection of the two circles C ..."
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Cited by 4 (1 self)
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We present an efficient and robust algorithm to compute the intersection curve of two ringed surfaces, each being the sweep ∪uC u generated by a moving circle. Given two ringed surfaces ∪uC u 1 and ∪vC v 2, we formulate the condition C u 1 ∩ Cv 2 �=∅(i.e., that the intersection of the two circles C u 1 and Cv 2 is nonempty) as a bivariate equation λ(u,v) = 0 of relatively low degree. Except for redundant solutions and degenerate cases, there is a rational map from each solution ofλ(u,v) = 0 to the intersection point C u 1 ∩ Cv 2. Thus it is trivial to construct the intersection curve once we have computed the zeroset of λ(u,v) = 0. We also analyze exceptional cases and consider how to construct the corresponding intersection curves. A similar approach produces an efficient algorithm for the intersection of a ringed surface and a ruled surface, which can play an important role in accelerating the raytracing of ringed surfaces. Surfaces of linear extrusion and surfaces of revolution reduce their respective intersection algorithms to simpler forms than those for ringed surfaces and ruled surfaces. In particular, the bivariate equation λ(u,v) = 0 is reduced to a decomposable form, f (u) = g(v)or�f(u) − g(v)�=r(u), which can be solved more efficiently than the
On Approximation in Spaces of Geometric Objects
 Mathematics of Surfaces IX
, 2000
"... this paper, we will restrict the class of developable surfaces we are working with: We only consider surfaces whose family of tangent planes is of the form U(t) = (u 0 (t); u 1 (t); u 2 (t); 1) () z = u 0 (t) + u 1 (t)x + u 2 (t)y: (17) 10 Helmut Pottmann and Martin Peternell For NURBS surfaces th ..."
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Cited by 4 (1 self)
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this paper, we will restrict the class of developable surfaces we are working with: We only consider surfaces whose family of tangent planes is of the form U(t) = (u 0 (t); u 1 (t); u 2 (t); 1) () z = u 0 (t) + u 1 (t)x + u 2 (t)y: (17) 10 Helmut Pottmann and Martin Peternell For NURBS surfaces this is equivalent to the choice of control planes U i = (u 0;i , u 1;i , u 2;i , u 3;i ) such that always u 3;i = 1. This means that for all possible planes U we no longer allow to choose an arbitrary coordinate quadruple describing U , but we restrict ourselves to the unique one whose last coordinate equals 1. This is not possible if the last coordinate is zero, so we have to exclude all surfaces with tangent planes parallel to the zaxis. In most cases this requirement is easily fullled by choosing an appropriate coordinate system
RATIONAL OFFSET SURFACES AND THEIR MODELING APPLICATIONS
"... Abstract. This survey discusses rational surfaces with rational offset surfaces in Euclidean 3space. These surfaces can be characterized by possessing a field of rational unit normal vectors, and are called Pythagorean normal surfaces. The procedure of offsetting curves and surfaces is present in m ..."
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Cited by 3 (1 self)
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Abstract. This survey discusses rational surfaces with rational offset surfaces in Euclidean 3space. These surfaces can be characterized by possessing a field of rational unit normal vectors, and are called Pythagorean normal surfaces. The procedure of offsetting curves and surfaces is present in most modern 3dmodeling tools. Since piecewise polynomial and rational surfaces are the standard representation of parameterized surfaces in CAD systems, the rationality of offset surfaces plays an important role in geometric modeling. Simple examples show that considering surfaces as envelopes of their tangent planes is most fruitful in this context. The concept of Laguerre geometry combined with universal rational parametrizations helps to treat several different results in a uniform way. The rationality of the offsets of rational pipe surfaces, ruled surfaces and quadrics are a specialization of a result about the envelopes of oneparameter families of cones of revolution. Moreover a couple of new results are proved: the rationality of the envelope of a quadratic twoparameter family of spheres and the characterization of classes of Pythagorean normal surfaces of low parametrization degree.
Points on Algebraic Curves and the Parametrization Problem
 In Automated Deduction in
, 1997
"... A plane algebraic curve is given as the zeros of a bivariate polynomial. However, this implicit representation is badly suited for many applications, for instance in computer aided geometric design. What we want in many of these applications is a rational parametrization of an algebraic curve. There ..."
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Cited by 3 (2 self)
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A plane algebraic curve is given as the zeros of a bivariate polynomial. However, this implicit representation is badly suited for many applications, for instance in computer aided geometric design. What we want in many of these applications is a rational parametrization of an algebraic curve. There are several approaches to deciding whether an algebraic curve is rationally parametrizable and if so computing such a parametrization. In all these approaches we ultimately need some simple points on the curve. The field in which we can find such points crucially influences the coefficients in the resulting parametrization. We show how to find simple points over some practically interesting fields. Consequently, we are able to decide whether an algebraic curve defined over the rational numbers can be parametrized over the rationals or the reals. Some of these ideas also apply to parametrization of surfaces. If in the term geometric reasoning we do not only include the process of proving or disproving geometric statements, but also the analysis and manipulation of geometric objects, then algorithms for parametrization play an important role in this wider view of geometric
Envelopes  Computational Theory and Applications
"... Based on classical geometric concepts we discuss the computational geometry of envelopes. The main focus is on envelopes of planes and natural quadrics. There, it turns out that projective duality and sphere geometry are powerful tools for developing efficient algorithms. The general concepts are il ..."
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Cited by 2 (0 self)
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Based on classical geometric concepts we discuss the computational geometry of envelopes. The main focus is on envelopes of planes and natural quadrics. There, it turns out that projective duality and sphere geometry are powerful tools for developing efficient algorithms. The general concepts are illustrated at hand of applications in geometric modeling. Those include the design and NURBS representation of developable surfaces, canal surfaces and offsets. Moreover, we present applications in NC machining, geometrical optics, geometric tolerancing and error analysis in CAD constructions.